Let F be a p-adic field, E be a quadratic extension of F, D be an F-central division algebra of odd index and let θ be the Galois involution attached to $E/F$. Set $H=\mathrm{GL}(m,D)$, $G=\mathrm{GL}(m,D{\otimes }_{F}E)$, and let $P=MU$ be a standard parabolic subgroup of G. Let w be a Weyl involution stabilizing M and $M^{{\theta }_w}$ be the subgroup of M fixed by the involution ${\theta }_w:m\mapsto \theta (wmw)$. We denote by $X{(M)}^{w,-}$ the complex torus of w-anti-invariant unramified characters of M. Following the global methods of [30], we associate to a finite length representation σ of M and to a linear form $L\in {\mathrm{Hom}}_{M^{{\theta }_w}}(\sigma ,\mathbb{C})$ a family of H-invariant linear forms called intertwining periods on ${\mathrm{Ind}}_P^G(\chi \sigma )$ for $\chi \in X{(M)}^{w,-}$, which is meromorphic in the variable χ. Then we give sufficient conditions for some of these intertwining periods, namely the open intertwining periods studied in [13], to have singularities. By a local/global method, we also compute in terms of Asai gamma factors the proportionality constants involved in their functional equations with respect to certain intertwining operators. As a consequence, we classify distinguished unitary and ladder representations of G, extending respectively the results of [42] and [26] for $D=F$, which both relied at some crucial step on the theory of Bernstein-Zelevinsky derivatives. We make use of one of the main results of [12] which in the case of the group G asserts that the Jacquet-Langlands correspondence preserves distinction. Such a result is for essentially square-integrable representations, but our method in fact allows us to use it only for cuspidal representations of G.

设F为p进场，E为F的二次扩展，D为奇数指数的F中心除代数，设θ为附加到$乙/F$. 放$H=\mathrm{GL}(米,D)$, $G=\mathrm{GL}(米,D{\otimes }_F乙)$， 然后让 $磷=米你$是G的标准抛物线子群。设w是稳定M的 Weyl 对合，并且${米}^{{\theta }_{瓦}}$是由对合确定的M的子群${\theta }_{瓦}：米\mapsto \theta (瓦米瓦)$. 我们表示为$X{(米)}^{瓦,-}$M的w -反不变非分支特征的复环面。以下的全局方法[30]，我们关联到有限长度表示σ的中号和线性形式$升\in {\mathrm{坎}}_{{米}^{{\theta }_{瓦}}}(\sigma ,\mathbb{C})$一族H不变的线性形式，称为交织周期${\mathrm{工业}}_{磷}^G(\chi \sigma )$ 为了 $\chi \in X{(米)}^{瓦,-}$，它在变量χ 中是亚纯的。然后我们给出了其中一些交织期，即[13]中研究的开放交织期，具有奇点的充分条件。通过局部/全局方法，我们还根据 Asai 伽马因子计算了其函数方程中涉及的关于某些交织算子的比例常数。因此，我们对G 的区别幺正和阶梯表示进行分类，分别扩展了 [42] 和 [26] 的结果，用于$D=F$，这两者都依赖于 Bernstein-Zelevinsky 导数理论的一些关键步骤。我们利用 [12] 的主要结果之一，在G组的情况下，该结果断言 Jacquet-Langlands 对应保留了区别。这样的结果基本上适用于平方可积表示，但我们的方法实际上允许我们仅将其用于G 的尖点表示。